# Dragon Notes

UNDER CONSTRUCTION
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# Linear Systems:Special Functions

\ds\boxed{\delta(t) = \left\{ \begin{align} &1,\ t = 0 \\ &0,\ \t{otherwise}\end{align}\right.}
Dirac-delta (Impulse)

• $$\ds \ilim{\Omega}\frac{\t{sin}(\Omega t)}{\pi t}=\delta(t),\ \sfrac{1}{2\pi}\pnint\t{cos}(\omega t)d\omega = \delta(t)$$
• $$\ds \sfrac{1}{2\pi}\pnint\t{sin}(\omega t)d\omega = 0,\ \sfrac{1}{2\pi}\pnint e^{-j\omega t}d\omega = \delta(t)$$
• $$\ds \ilim{\omega}\pnint\t{sin}(\omega t)\phi(t)dt = 0,\ \ilim{\omega}\pnint\t{cos}(\omega t)\phi(t)dt = 0,\ \forall \phi(t)$$
• $$\ds \pnint f(t)dt=1 \Ra \lim_{a \ra 0} \sfrac{1}{a}f\lrpar{\sfrac{t}{a}} = \delta(t),\ \forall f(t)$$